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Properties

Label 2-1407-1407.950-c0-0-0
Degree $2$
Conductor $1407$
Sign $-0.669 - 0.742i$
Analytic cond. $0.702184$
Root an. cond. $0.837964$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.981 − 0.189i)3-s + (−0.235 + 0.971i)4-s + (−0.142 + 0.989i)7-s + (0.928 + 0.371i)9-s + (0.415 − 0.909i)12-s + (0.165 + 0.231i)13-s + (−0.888 − 0.458i)16-s + (0.140 + 0.351i)19-s + (0.327 − 0.945i)21-s + (−0.995 + 0.0950i)25-s + (−0.841 − 0.540i)27-s + (−0.928 − 0.371i)28-s + (−1.67 − 0.159i)31-s + (−0.580 + 0.814i)36-s + (0.995 + 1.72i)37-s + ⋯
L(s)  = 1  + (−0.981 − 0.189i)3-s + (−0.235 + 0.971i)4-s + (−0.142 + 0.989i)7-s + (0.928 + 0.371i)9-s + (0.415 − 0.909i)12-s + (0.165 + 0.231i)13-s + (−0.888 − 0.458i)16-s + (0.140 + 0.351i)19-s + (0.327 − 0.945i)21-s + (−0.995 + 0.0950i)25-s + (−0.841 − 0.540i)27-s + (−0.928 − 0.371i)28-s + (−1.67 − 0.159i)31-s + (−0.580 + 0.814i)36-s + (0.995 + 1.72i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1407 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1407\)    =    \(3 \cdot 7 \cdot 67\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(0.702184\)
Root analytic conductor: \(0.837964\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1407} (950, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1407,\ (\ :0),\ -0.669 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5420720677\)
\(L(\frac12)\) \(\approx\) \(0.5420720677\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.981 + 0.189i)T \)
7 \( 1 + (0.142 - 0.989i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
good2 \( 1 + (0.235 - 0.971i)T^{2} \)
5 \( 1 + (0.995 - 0.0950i)T^{2} \)
11 \( 1 + (-0.995 + 0.0950i)T^{2} \)
13 \( 1 + (-0.165 - 0.231i)T + (-0.327 + 0.945i)T^{2} \)
17 \( 1 + (0.841 + 0.540i)T^{2} \)
19 \( 1 + (-0.140 - 0.351i)T + (-0.723 + 0.690i)T^{2} \)
23 \( 1 + (-0.928 - 0.371i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (1.67 + 0.159i)T + (0.981 + 0.189i)T^{2} \)
37 \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.888 - 0.458i)T^{2} \)
43 \( 1 + (0.512 - 1.74i)T + (-0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.142 - 0.989i)T^{2} \)
53 \( 1 + (0.0475 + 0.998i)T^{2} \)
59 \( 1 + (-0.327 - 0.945i)T^{2} \)
61 \( 1 + (-0.0934 + 1.96i)T + (-0.995 - 0.0950i)T^{2} \)
71 \( 1 + (-0.0475 - 0.998i)T^{2} \)
73 \( 1 + (0.495 - 0.770i)T + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (-0.172 - 0.0789i)T + (0.654 + 0.755i)T^{2} \)
83 \( 1 + (-0.995 + 0.0950i)T^{2} \)
89 \( 1 + (0.928 - 0.371i)T^{2} \)
97 \( 1 + (-0.327 + 0.566i)T + (-0.5 - 0.866i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.879686775852611384004769122298, −9.340080580246237619939450766132, −8.292194541576520594028698804766, −7.69260769076020284443112629436, −6.70599798642738474404302320462, −5.96510142393842088577390912253, −5.09036129881049967491543833732, −4.18480624437769774251176453300, −3.11060027993320054524501481491, −1.83591964778193283485430241604, 0.50954482956888653242004690697, 1.81195888455005542476241577883, 3.72817697638937307504379121824, 4.43471626603974404066719408807, 5.44216271944935448434469148436, 5.95722381551466552914045974295, 6.95719636515083584287068353513, 7.53903828882820062052489618476, 8.960996606736394480268026892452, 9.644878058851933656286090283506

Graph of the $Z$-function along the critical line